Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

0
votes
0answers
26 views

Is the standard model of the reals the only model up to isomorphism of cardinality $\beth_1$? [duplicate]

TL;DR: I'm trying to figure out what you need to do to pin down the standard model of the reals without explicitly constructing such a model and then defining the standard model $W$ of the reals up to ...
1
vote
0answers
21 views

Finite sets of a infinite set A is in bijection with A [duplicate]

Suppose $A$ is a infinite set. Let $P_f(A)$ be the collection of all finite subsets of $A$. Now we want to show that $P_f(A)$ is in bejection with $A$. I can do this for the case that $A$ is countable,...
3
votes
2answers
40 views

Cardinality of the set of all total orders on $\Bbb{N}$

I need to compute the cardinality of the set of all total orders on $\Bbb{N}$. Now, by definition there is an inclusion of this set into $\mathcal{P}(\Bbb{N}\times\Bbb{N})$, and so has cardinality $\...
3
votes
2answers
39 views

Countability of a subset of sequences

Let $\mathcal{A} = \{a \in \{1,2,3,4,5\}^\Bbb N : |a_i- a_{i+1}| = 1 \; \forall i\}.$ Is the set $\mathcal{A}$ countable? I tried an argument like Cantor's diagonalization process but without success....
2
votes
1answer
41 views

Cardinality of all infinite subsets with an infinite complement.

Working in $\text{ZF}$... Let $X$ be an infinite set with a given well-ordering relation $\le$. Define $\tag 1 \mathcal B(X) = \{ S \in \mathcal P(X) \, | \, S \text{ is infinite } \text{ and } X \...
2
votes
2answers
62 views

$\aleph_0\times 2^{\aleph_0} = 2^{\aleph_0}$ requires the Axiom of Choice?

In a model solution it is stated that the cardinality of a set which is the countable union of sets of cardinality $2^{\aleph_0}$ is $\aleph_0\times 2^{\aleph_0}$, and, using the Axiom of Choice, $\...
0
votes
1answer
33 views

find cardinality of A [duplicate]

$A$ = $\{X\subseteq \mathbb N : |X| = \aleph_0\land |\bar X| \lt\aleph_0 \} $ I need to find the cardinality of group A. As I understood, I need to find a function that helps me determine the ...
0
votes
1answer
53 views

Which stage in the Neumann hierarchy do powers of the reals fit in?

To be more specific than the short title, I try to gauge the size of some "normal" function spaces as e.g. found in functional analysis against set universe sizes at certain stages. For the sake of ...
0
votes
0answers
22 views

what is an Injective function from A to B when A=$P(\mathbb N)$ B=$\{T⊆N|∀S⊆T.|T|<א_0→\sum_{x∈S} x≠\sum_{x∈T/S} x\}$?

what is an Injective function from A to B when A=$P(\mathbb N)$ B=$\{T⊆N|∀S⊆T.|T|<א_0→\sum_{x∈S} x≠\sum_{x∈T/S} x\}$ ?
0
votes
1answer
32 views

Cardinal of all well-orders of $\mathbb{N}$ [duplicate]

Lets consider the set $$ A = \{R\subset\mathbb{N}^2:R \text{ is a well-order of } \mathbb{N}\} $$ Now, it's clear that $\aleph_1\leq|A|\leq2^{\aleph_0}$(Since all the well-orders of $\mathbb{N}$ are ...
2
votes
1answer
20 views

Does a (complete) $\mathfrak{a}$-partite graph always have a maximum in AND out degree?

A graph $G$ is called $\mathfrak{a}$-partite (with $\mathfrak{a}$ being any non empty cardinal), if there exists a partition $\mathcal P$ of $V(G)$ such that any two vertices in the same class $A\in\...
2
votes
0answers
79 views

Show that continuous function from omega_1 to R is eventually constant [duplicate]

Let $f: \omega_1 \to \mathbb{R}$ be a contionuous function. Prove that $f$ is eventually constant. I was trying to prove it by contradiction but I did not have any idea.
1
vote
0answers
29 views

Why is cofinite topology on a countable set a second countable space? [duplicate]

Consider a countable set $X$ with cofinite topology. Here the claim is that $X$ is second countable. The simple argument is as follows : $\color{red}{\text{Since $X$ is countable, the number of ...
3
votes
1answer
40 views

Are there uncountable cardinals $\kappa$ such that $|\kappa\cap\mathsf{Card}| = \kappa$?

All the cardinals $\kappa\leq\aleph_0$ have the property that there are precicely $\kappa$ cardinals less than $\kappa$. Of course, $\aleph_1$ lacks this property since there are only $\aleph_0 +1= \...
1
vote
2answers
93 views

What is the analogue to this cardinal arithmetic theorem for infinite products?

$\begin{array}{l}{\text { 1.3 Theorem Let } \lambda \text { be an infinite cardinal, let } \kappa_{\alpha}(\alpha<\lambda) \text { be nonzero cardinal}} \\ {\text {numbers, and let } \kappa=\sup \...
1
vote
0answers
36 views

cardinality of multiplication of groups A,B when $A\leq B$

if $A$ and $B$ are infinite groups and $$ |A|\leq|B|, $$ where $|{}\cdot{}|$ denotes the group cardinality, is it right to say that $|A|\cdot|B| = |B|$?
2
votes
1answer
42 views

How $c^{\aleph_0}=c$

I am reading about coutabaility, uncountability and cardinal numbers. I have attached herewith a screenshot of the wikipedia page. I am not able to understand how in the third equation we show that $...
2
votes
1answer
42 views

If $X$ is an infinite set and $N$ is a countable set, then what is the cardinality of $X \times N$? [duplicate]

If $X$ is countable, then the cartesian product is countable. However, what about general cases? The googling suggests that the answer is the cardinality of $X$. But why? Could anyone please provide ...
0
votes
1answer
20 views

what is the cardinality of equivalence classes of relation $ R=\{<A,B>\in P(\mathbb{N} )|A\cap T=B\cap T\} $?

given :$$T\subseteq \mathbb{N} $$ $$ R=\{<A,B>\in P(\mathbb{N} )|A\cap T=B\cap T\} $$ what is a equivalence relation what is the cardinality of equivalence classes of relation R ? how can I ...
2
votes
0answers
26 views

Binomial coefficients for infinite cardinalities [duplicate]

Let’s define $C_\alpha^\beta$ as the cardinality of the set of all subsets with cardinality $\beta$ of a set with cardinality $\alpha$: $$C_\alpha^\beta = |\{T \subset S| |T| = \beta \}|$$ where $|S|...
0
votes
2answers
30 views

List all the possible sizes of N. Which of these answers divides 60?

Suppose you have disjoint sets $A, B, C, D $, and $F$ with sizes 1, 12, 12, 15, and 20, respectively. Suppose N is a set formed by taking the union of A with one or more of the other sets. List all ...
1
vote
1answer
26 views

Cardinality of a union of ordinal-indexed nested sets with bounded cardinality?

Let $(X_\alpha)$ be a nested ordinal-indexed sequence of sets, i.e. if $\alpha\leq\beta$ then $X_\alpha\subseteq X_\beta$ for all ordinals $\alpha$ and $\beta$, and suppose that there is a cardinal $\...
1
vote
2answers
44 views

Proving that any two sets of lines that cover the plane have the same cardinality

Lets say that a set of lines of the real plane covers the plane if, for every element $\langle x,y\rangle\in\mathbb{R}^2$, there exists a line $l$ of the set that passes through $\langle x,y\rangle$. ...
3
votes
1answer
56 views

Why $|\Bbb{R}|=2^{\aleph_0}$ and not $|\Bbb{R}|=10^{\aleph_0}$?

Suppose we have a real number of this form: $$...x_3x_2x_1,x_{-1}x_{-2}x_{-3}...$$ Since $x_i\in\{0,1,2,3,4,5,6,7,8,9\} \ \ \ \forall i$. We have 10 choices for every number, this means that: $$|\...
1
vote
1answer
50 views

Calculate the cardinal of $\prod_{0<\alpha<\omega_1}\alpha$

My guess is that the cardinal $\prod_{0<\alpha<\omega_1}\alpha$ can be computed in the following way: $$\prod_{0<\alpha<\omega_1}\alpha=\aleph_0\prod_{\omega\le\alpha<\omega_1}\alpha=\...
4
votes
2answers
111 views

Is $\aleph_{1}^{\aleph_0}<\aleph_{2}^{\aleph_0}$ equivalent to the Continuum Hypothesis?

While searching for exercises dealing with CH and GCH, I encountered an exercise with the following statement: Study if: $\aleph_{1}^{\aleph_0}<\aleph_{2}^{\aleph_0}$ Is equivalent to ...
-4
votes
2answers
34 views

Value of Sets - Union and Intersection [closed]

I would like to know the best way to answer this question! Assume that $| A ∩ B |= 13$, $| A |= 17$ and $| B |= 19$. Determine the value of $| A ∪ B |$. Regards!
1
vote
2answers
37 views

Cartesian Product and Sets - Discrete Mathematics [closed]

I'm unsure on how to answer this type of question, please can someone explain how to answer these step by step: $1)$ Recall that the Cartesian product $A\times A$ is defined as the set $$\{(x,y):x\in ...
0
votes
2answers
69 views

Why is $\mathbb{R}$ unbounded, despite being equinumerous to various bounded sets? Is there a name for this “distinction”? [closed]

$[0, 1] \approx (0,1) \approx \mathbb{R}$, for example. Intuitively, it seems that the infinity of $\mathbb{R}$ is of a different nature than that of the intervals; with $\mathbb{R}$ I can “explode” ...
0
votes
0answers
26 views

about cardinal ,mapping on X [duplicate]

if A is a infinite set ,define B={f|f is 1-1 correspondence on A} I want to compute Card(B) My attempt when A is finite ,then card(B)=$2^{card(A)}-2$ Such as A={$a_1,a_2$},then there exist two ...
0
votes
0answers
28 views

Bijection between $[a,b)$ and $(a,b)$? [duplicate]

I know this question has been asked and answered before, but I am working on my own through an analysis textbook and just wanted to check if the following construction would be appropriate: Define $a|...
0
votes
0answers
32 views

Basic Proposition of Cardinals

Definition. A cardinal is an ordinal which it is not in bijection with any smaller ordinal. Notation. $|X|$ means that cardinal of $X$. Proposition 1. Let $w$ be an ordinal. Then $|w+1|=w.$ ...
2
votes
1answer
38 views

Counterexample for $\prod_{i<\nu} \kappa_i=(\sup_{i<\nu}\kappa_i)^{\nu}$ when $\kappa_i$ is not an increasing $\nu$-sequence of cardinals

It is known that $\displaystyle \prod_{i<\nu} \kappa_i=\left(\sup_{i<\nu} \kappa_i \right)^{\nu}$ if $\nu$ is an infinite cardinal and $\langle \kappa_i | i < \nu \rangle$ is an increasing $\...
0
votes
1answer
26 views

Calculate cardinal number of given A set

Let $x \in \mathbb R$. Let $\text A x = \{ y \in \mathbb R : |x-y| \in \mathbb Z \}$ Find $| \text A x |$. Now I understand I need to use a bijective function, to send $\text A x$ to a known ...
0
votes
0answers
34 views

For cardinals $a,b,b'$, if $a\ge 2$ and $b<b'$, then $a^b <a^{b'}$

I need to prove, without assuming the Axiom of Choice, that for cardinals $a,b,b'$, if $a\ge 2$ and $b<b'$, then $a^b <a^{b'}$. I have already proved that for cardinals $c,c',d,d'$, if $c\neq0$,...
0
votes
0answers
17 views

Proving the sum of two finite cardinal numbers is finite

I am trying to prove the following: If $m,n\in \omega$, then $m\oplus n<\omega$ My proof is as follows Say wolog $m\in n$ Let $$f:(\{0\}×m)\cup (\{1\}×n)\rightarrow 2×\omega$$ $$f(t, p)=<...
1
vote
1answer
52 views

Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection [duplicate]

Question: Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection. From what I've read about infinite families, I need to ignore those who have the properpty $...
0
votes
1answer
47 views

Indexing of uncountable sets and uncountable collections of sets, uncountable intersections containing a point

Definitions Let $\mathcal{A}$ be an uncountable collection of sets so that if $I_{\mathcal{A}}$ is the index set of elements of $\mathcal{A}$ then $|I_{\mathcal{A}}|\not=\aleph_{0}$ (I mean this to ...
1
vote
0answers
27 views

How to fill a rectangle with smaller specific rectangles that have cardinal information about their adjacent neighbours

Lets say its 6x6 grid that is represented by top left(0,0) and bottom right(1,1) in coordinate system. Next, I have set of objects with their cardinal directional information about each of their ...
-1
votes
2answers
50 views

The cardinality of all equivalence relations over $\mathbb{N}$ [duplicate]

Let $R$ be the set the contains all equivalence relations over $\mathbb{N}$. Prove that $\left | R \right | = 2^{\aleph_0}$ This question is very counter - intuitive to me. I know that Each $R_i \...
1
vote
1answer
51 views

Finiteness of sets

For each of the following sets, determine if it is finite, countably infinite, or uncountable. You need not prove your answer is correct, but you should give a reason for your response. For some $n\...
0
votes
1answer
47 views

Countability of decimal representations of real numbers

Let $X=\{x\in \mathbb{R}\ | \ \hbox{the decimal representation of $x$ contains only 4s and 7s}\}$. Is $X$ countable or uncountable? Prove that your answer is correct. Should an argument like Cantor's ...
0
votes
0answers
39 views

Cardinality of partitions of a number

A partition of $n$ is a sequence of integers $(a_1, a_2, \dots, a_k)$ such that $a_i\geq 0$ for each $i$, and $\displaystyle\sum_{i=1}^k a_i = n$. The number $k$ is called the number of parts of the ...
1
vote
2answers
45 views

Cardinality of an arbitrary interval of real numbers

Let $a, b\in\mathbb{R}$ with $a<b$. Prove that $|\{x\in \mathbb{R}\ | \ a< x< b\}|=|\{x\in \mathbb{R}\ | \ 0<x<1\}|$. Would constructing a bijection be the most effective way to prove ...
0
votes
1answer
18 views

to show cardinality in separable Hilbert space setting

enter image description here the above exercise in Conway's Functional Analysis book to me, the setting is too rough so i have no idea to step forward could you help me to start this proof? or just ...
2
votes
0answers
40 views

Functions between ordinals.

I'm trying to compute the cardinality of a determined set. $$A=\{f\colon \omega_n \to \omega_m | |Supp(f)|=\aleph_k \quad k<n\}$$ As suggested by the exercise, I first tried few elementar cases: $$...
0
votes
1answer
37 views

Aronszajn tree for infinite singular cardinal [duplicate]

I've always seen Aronszajn trees being discussed on regular cardinals : Let $\kappa$ be a regular infinite cardinal. A $\kappa$-Aronszajn tree is a tree on $\kappa$ of height $\kappa$, whose ...
2
votes
1answer
22 views

Bijection between $ \bigcup_{i \in [0, 1]} X_i \ \ \ \text{and} \ \ \ [0, 1] \times [0, 1] $

So I have got the following sets $$ \bigcup_{i \in [0, 1]} X_i \ \ \ \text{and} \ \ \ [0, 1] \times [0, 1] $$ where each $X_i$ has cardinality $c$ of the continuum and each pair of $X_i$ where $i \in [...
0
votes
1answer
28 views

$X$ is defined as the collection of sets $X = \{X_i: i \in [0, 1] \}$ where each set has cardinality $c$

$X$ is defined as the collection of sets $X = \{X_i: i \in [0, 1] \}$ where each set has cardinality $c$ of the continuum and each pair is disjoint. How do I prove that for each $i \in [0, 1]$, the ...
0
votes
0answers
57 views

An uncountable set has uncountably many co-countable subsets containing an arbitrary point

This theorem seems "obvious" to me, but I want to check my logic since I am un-familiar with un-countably infinite sets, and I know these can give rise to non-intuitive results. Any comments welcome. ...